Dispersion total photoproduction sum rules for nucleons and few-body nuclei revisited

0.12 MB·English

by  S. B. Gerasimov

Dispersion total photoproduction sum rules for nucleons and
few-body nuclei revisited
S.B. Gerasimov∗
Joint Institute for Nuclear Research,141980 Dubna, Russia
Abstract
The questions on the presense and quantitative role of the constant terms in the real part of
the high-energy photon-nucleon and photon-nucleus amplitudes representing the contribution of
the non-Regge (the fixed j = 0-pole) singularities in the finite-energy sum rules (FESR) for the
photoabsorption cross sections on nucleons and the lightest atomic nuclei are discussed and new
testable relations arepresentedfor relevant combinations of theComptonscattering amplitudes.
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Electronic address: [email protected]
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I. INTRODUCTION
In the 1954 seminal paper of Gell-Mann, Goldberger and Thirring (GGT)[1] on the
use of the causality condition in quantum theory the idea of the ”superconvergence” sum
rule technique was first suggested and applied to the photonuclear absorption processes.
The GGT sum rule follows from the assumption of validity of the unsubtracted dispersion
relations for the difference presumably vanishing as ν → ∞
∆T = T (ν)−ZT (ν)−NT (ν) (1)
γA γp γn
of the forward Compton scattering amplitudes on the nucleus with atomic number A =
Z + N and the sum of amplitudes on the Z free protons and N free neutrons. After
inclusion of the Thompson value −(αQ2)/M (M(Q) being the hadron mass and electric
charge in the units of the electron charge), for every hadron amplitude T(ν = 0) at zero
photon energy, the sum rule reads
α −Z2 ∞ νγπ
2π2 ( +Z)+ dν[Zσ (ν)+Nσ (ν)−σ (ν)] = dνσ (ν) (2)
M A Z γp γn γA Z γA
n νγπ νthr
The first term in l.h.s. of (2) practically coincides with the ”kinetic” part of the long-
known Thomas-Reiche-Kuhn sum rule for the electric dipole nuclear photoabsorption
∞
σ (E1) ≡ σ (ν)dν = 4π2Σ (E −E )|hn|D |Ai|2 = 2π2hA|[D [H,D ]]|Ai
0 Z E1 n n 0 z z z
νthr
= (2π2αNZ)/(AM )+2π2hA|[D [Vˆ ,D ]]|Ai (3)
n z NN z
where the first term results from the double commutator with the kinetic energy operator
of the nuclear hamiltonian H. The present work originates partly from earlier papers
of the author [2, 3, 4] dealing with sum rules for total photon-hadron cross-sections and
aims to present some new experimentally testable and theoretically interesting relations
emerging from the dispersion FESR phenomenology.
II. TOWARDS GENERALIZED GGT SUM RULE
It was always tempting and rewarding to combine the power of dispersion relation ap-
proach, which is based on very general underlying assumptions and explains many general
properties of the scattering amplitudes as well as provides useful relations between them
in a rather simple way with particular dynamical ingredients of a given quantum system
such as, for instance, implications of the broken chiral symmetry and pionic dynamics
dominating peripheral properties and low-energy interactions of hadrons and nuclei.
A. A glance at a possible role of pion degrees of freedom on GGT sum rule
In[2,4]anattemptwasmaidtointroducecorrectionstotheGGTapproachunderstood
as a familiar Impulse Approximation (IA) scheme applied to the γA -forward scattering
2
amplitude. The approximate relevance of IA is seen from the fact that it corresponds
to taking into account the singularities closest to the physical region of the peripheral
scattering process (t ≤ 0, t = (k −k′)2 is the invariant 4-momentum transfer for elastic
scattering). The respective cut inthe complex t-planeis defined by thediagramsschemat-
ically represented in Fig.1a, while the next to the leading ”anomalous” threshold given
by Fig.1a will be the ”normal” 2π -exchange diagrams, represented in Fig.1b, with the
cut starting at t = 4m2.
π
Fig.1
∼∼∼ (cid:13) ∼∼∼ ∼∼∼ (cid:13) ∼∼∼
. .
. .
րց . .
(a) =⇒ (cid:13) −→ (cid:13) =⇒ (b) =⇒ (cid:13) =⇒
In Fig.1, the solid lines refer to nucleons and nuclei, the wavy (dotted) lines represent
photonsandpions. Graph(a)represents theimpulseapproximation(IA),while(b)defines
the correction related with the nuclear ”collective” pion cloud and thus is effective due
to short-ranged NN-correlation inside nuclei. Their relative role can qualitatively be
characterized by the ratio
t (IA) 8m ε
0 n b
≃ (4)
t (2π) (A−1)·4m2
0 π
where t refers to the beginning of the respective cut in the complex t-plane, m is the
0 n
nucleon mass, ε is the nuclear binding energy. For instance, this ratio is ∼ .22 (.40 and
b
.66) for the d (3He and 4He, respectively). This indicates that, naturally, for 3He and
4Hethe”pionic”contributionswillbesignificantlymoreimportantcomparedtodeuteron.
Equation (4) also signals that, in the considered respect, the situation for heavier nuclei
is expected to be much alike the 4He case because of nearly equal binding energy per
nucleon.
B. Towards the measurement and systematization of < A|φ∗φ|A >
A further step in relevant implementation of pionic d.o.f. into the GGT sum rule was
an observation inferred from models providing the convergence of the σ (tot)-integral. It
0
was first suggested [3] and then perturbatively (to one-loop order) checked [5] in scalar,
φ3-type ”super-renormalizable” model that the generalized Thomas-Reiche-Kuhn is valid
for total photoabsorption cross section
σ = dνσ (ν) = 2π2hφ |[D[H,D]]|φ i, (5)
0 Z tot 1 1
where the charged scalar field φ is locally connected with two scalar fields, φ being
1 2
charged one and the other, φ ,neutral. The double commutator is then interpreted via
3
the known Schwinger-term, i.e., the equal-time commutator of the time- and spatial-
′
component of e.m. current operator. Hence, the generalized, ”GGT ”-sum rule, implicitly
3
including theintegralsoftheabsorptive partsoftheamplitudes presented bythediagrams
with 2π-exchanges,was written [4] in the form
NZ
σγA −Zσγp −Nσγn = 2π2α[ + d~x(hA|φ∗φ|Ai−Σ hN |φ∗(x)φ(x)|N i)]. (6)
0 0 0 Am Z i i i
The photonuclear sum rule including the terms < A|φ∗φ|A > and < N|φ∗φ|N >, repre-
sented by the Feynman diagram in Fig.2
Fig.2
∼∼∼ • ∼∼∼
. .
. .
. .
=⇒ (cid:13) =⇒
was later rediscovered [6], found to be a useful exploration tool [7] and widely discussed
(e.g., [8] and further references therein) in view of the interesting idea about possible
partial restoration of the chiral symmetry in real nuclei.
The matter is that up to constant factors the matrix element corresponding to the
seagull graph in Fig.2, which in the forward direction is gauge invariant and may have a
direct bearing to measurable quantities, is essentially of the same structure as the matrix
element represented by Fig.3
Fig.3
⊗
. .
. .
. .
=⇒ (cid:13) =⇒
Thesymbol ⊗denotes thelocal, scalar quark-current andthereforeit isdirectly connected
with the Σ -term, hence with the chiral symmetry breaking and its possible (partial)
π
restoration in nuclear matter.
C. FESR and problem of ”Big Circle” contribution
The standard FESR technique enables one to deal with the amplitudes defined in the
finite region of the complex energy plane
1 f(z)
f(ν) = dz (7)
2πı I z −ν
where f(ν) is the spin-averaged, forward Compton scattering amplitude and the integra-
tion contour includes both sides of the cuts along the real axes −R ≤ ν ≤ R closed by a
circle of a ”big” radius R. As usual the problem consists in the justified and economical
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choice for the representation of amplitudes in the complex energy plane to fulfill the inte-
gration over the large but finite-radius circle in the complex plane. We keep the original
GGT idea of a relation between the photon-nucleus scattering amplitude and a relevant
combination of the photon-nucleon amplitudes at sufficiently large photon energies, but
our choice of the ”superconvergent” combination of Compton amplitudes f is differ-
γA(p,n)
ent from GGT.It includes amplitudes of two nuclei with A = Z +N ,A = Z +N and
1 1 1 2 2 2
is assumed to satisfy the condition:
1 1 Z N −N Z S (A ) S (A )
1 2 1 2 π 1 π 2
lim [ f − f ] = (f −f )| + − (8)
|ν|→R A A1 A A2 A A p n ν=R A A
1 2 1 2 1 2
where
α
S (A ) ≃ d3xhA |φ~(x)φ~(x)|A i (9)
π i 3 Z i i
and the scalar product in the integrand is understood to be in the isospin space.
The upper limit ν ≡ R in all integrals should be chosen from the compromise provi-
max
sions.
The first term derived in the approximation linear in A , (i = 1,2) is parameterized
i
through the a (JP;IG = 2+;1−) -Reggeon exchange in the t-channel and includes in
2
addition the real constant term seemingly taking place [9] in the Ref and referring as
p
the residue of the j = 0 fixed-pole in the complex angular momentum plane. Hence one
should put R ≥ 1.5÷2.0 GeVto applythe Regge-polephenomenology with thecommonly
used parameters [10]
Im[f (ν)−f (ν)] = (ν/4π)(σtot −σtot) = b ν1/2
p n p n a2
Re(f (ν)−f (ν)) = (1/4π)b (−ν1/2)+C −C
p n a2 p n
σtot(ν)−σtot(ν) = 24.6/ν1/2 (10)
p n
Following [9], we accept C ≃ −3.0µb·GeV and put the C -value rather arbitrarily to be
p n
either C = (2/3)C or C = 0 for the sake of further numerical estimations.
n p n
Due to the dominant scalar-isoscalar nature of the pionic operators we accept S (p) ≃
π
S (n) while S (A ) 6= 0 will disclose its essential nonlinear dependence on the atomic
π π i
number A of real nuclei.
III. NUMERICAL RESULTS AND DISCUSSION
As an example of the generalized nuclear sum rule applications, we choose a pair of
lightest nuclei - the deuteron and 3He. While in the deuteron case the total photoab-
sorption cross section is known well above our taken ν ≃ 1.6 GeV, the σ (γ3He) is
max tot
known to .8 GeV [? ] ; hence, in this case, we have to take ν = .8 GeV. The major
max
purpose of using these new types nuclear sum rules may be the extraction of informa-
tion about the value of difference of the nuclear matrix elements: ∆σ = d~xm2π[ 1 <
π 2 A1
R
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A |φ~(x)·φ~(x)|A > − 1 < A |φ~(x)·φ~(x)|A >].
1 1 A2 2 2
The term ∆σ can thus be extracted from experimentally measurable quantities to give
π
useful information on the values closely related with the chiral symmetry characteristics
in real nuclei. Of special interest is the situation when Z N −N Z = 0 in (8), as for the
1 2 1 2
deuteron- and 4He-pair, to mention. The contribution of the a -Reggeon is then absent
2
and the optimal value of ν = R in dispersion integrals of cross sections could probably
max
be taken at a lower value. Qualitatively, this newly chosen R-value should provide a rea-
sonable balance between the contribution of the same group of most important nucleon
resonances into the real parts of nuclear Compton amplitudes represented by the terms
S (A ) and the respective imaginary parts entering into dispersion integrals in the form
π i
of the corresponding nuclear photo-pion production cross sections.
For arbitrary A = Z +N and A = Z +N our general sum rule reads
1 1 1 2 2 2
f (ν = 0)+S (A ) f (ν = 0)+S (A )
2π2[ A1 π 1 − A2 π 2 +
A A
1 2
Z N −Z N 2b ν1/2 σνmax(γA ) σνmax(γA )
+ 1 2 2 1 ·( a2 max −C +C )] = 0 1 − 0 2 (11)
A A 2π2 p n A A
1 2 1 2
where f (ν = 0) ≃ −(αZ2)/(A m ) is the Thompson zero-energy amplitude, S (A )
Ai i i n π i
is defined in Eq.(9) and the integration in σνmax extends from the photodisintegration
0
threshold to the upper bound ν . In the case of 3He and deuteron the integration was
max
carried out with the cross-sections tabulated in [11] up to ν = .8 GeV. The low-energy
max
integrals up to the pion photoproduction thresholds ν were approximated by
γπ
NZ
σνγπ = 60 (1+κexp) [µb·GeV] (12)
0 A A
where κexp = .75±.10 (.37±.11), following [12]. To have an idea about the scale of
He−3(d)
S for the nuclei considered, we confronted the calculated values of
π
(2π2α)/(3)[(1/3)S (3He)−(1/2)S (d)] ≃ 7.75 (1.17) [µb·GeV]
π π
for C = −3,C = −2 (0) with the value
p n
60·[(2/9)κ3He−(1/4)κd] = (1/3)·40·(.75±.10)−(1/2)·30·(.37±.11) ≃ 4.4±2.1 [µb·GeV]
representing the ”potential parts” in the difference of non-relativistic TRK sum rules
2π2α[(1/3) <3 He|[D,[V ]]|3He > −(1/2) < d|[D,[V ]]|d >]
NN NN
The correspondence looks reasonable because the non-relativistic value is in between two
valuesfollowingfromamoregeneralsumrulewiththedifferingvaluesofC . Wealsodraw
n
attention to a strong dependence of the mentioned estimations on two chosen numerical
values of C , which emphasizes the significance of sum rule as a source of new interesting
n
information.
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IV. CONCLUSION
The pion-nucleon sigma-term
mˆ 1
¯
σ = hp|u¯u+dd|pi, mˆ = (m +m ),
u d
2m 2
p
and, generally, sigma-termsofagivenhadronareproportionaltothescalar quarkcurrents
hA|m q¯q|Ai ;q = u,d,s ; A = π,K,N, ZA −nuclei.
q N
These are of great physical significance because they are related to the hadron masses, to
themesonscatteringamplitudes[13],tothestrangenesscontentofA,andtotheproperties
ofnuclear [14]anddark[15]matter. Ourderived anddiscussed photoabsorptionsumrules
are focused on the comparison of the scalar pion densities, hence, in part, of the pionic σ-
terms for different nuclei to trace their dependence on the atomic number. The deuteron
sum rule provides thereupon the situation most close to free nucleons while the helium-4
plays the role of a drop of the real nuclear matter.
In view of the above discussion the following looks to be practically important:
1. To extend measurements of the total photoabsorption on the 3He and 4He -nuclei at
least up to energy of photons 1.5÷2.0 GeV.
2. To complete calculation of < A|[D[H,D]]|A >, A =3(4) He, with best modern po-
tentials andrespective wave functions aswell as with estimation of relativistic corrections.
Partial support of this work by the Blokhintsev-Votruba foundation is gratefully acknowl-
edged.
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